A system for constructing relatively small polyhedra from Sonobe modules

This article assumes a basic familiarity with the Sonobé module, including the ability to construct an augmented icosahedron. [3] (pp. 42ff) and [5] each contain excellent introductions 2 Often erroneously called "stellated". A polyhedron is augmented by capping each face with a pyramid. Look at the octahedron again. Notice that each of the original vertices not only has four pyramids meeting at it but also has eight edges meeting at it. Of these eight edges, four are "mountain" edges, which go to the tip of one of the pyramids, and four are "valley" edges, which go to another one of the original vertices in the diagram. One can construct the octahedron from the diagram by choosing an original vertex to start at, putting four Sonobés in a cycle to represent the four valley edges, and then adding four more modules in the usual manner to make four pyramids surrounding the original vertex. Then choose one of the four original vertices connected to the starting vertex by an edge in the diagram, consult the diagram to see that it is meant to be surrounded by four pyramids, add sufficient Sonobés in a cycle around that vertex to make four, and then complete the necessary pyramids. Continue in this manner until the model closes itself off, which will happen automatically. Now let's try the method with a new deltahedron: The pentagonal dipyramid. As far as I can ascertain, instructions for constructing an augmented version of this polyhedron out of Sonobé modules have never appeared either in print or on the internet 3 . Imagine a 5-sided pyramid made of equilateral triangles with a regular pentagon for a base. Now imagine sticking two of these together along their pentagonal bases to create a deltahedron with ten faces. It may be possible for you to imagine this clearly enough to see that the diagram of the original vertices of this deltahedron is: Now try constructing the augmented version with modules. Note that this deltahedron has ten faces. The number of modules necessary for constructing an augmented version of a deltahedron is always 3 2 times the number of faces in the original deltahedron, so that fifteen modules will be necessary. If you find it too difficult to imagine the polyhedron clearly enough to draw the above diagram it will probably be helpful to make a rough paper model of it first. Enlarge and copy the next diagram, cut it out and fold and tape it into a model of the pentagonal dipyramid (this kind of diagram is called a net of the polyhedron). Nets for all the convex deltahedra are available at [6], as well as rotatable pictures of some of the many nonconvex ones.

By means of this system it is possible to construct models of all the augmented convex deltahedra, as well as some others. One particularly interesting nonconvex deltahedron to construct is obtained from a cube by capping each of the six square faces with a pyramid which has four equilateral triangular sides. Instructions for making a beautiful semi-skeletal model of this 24-sided deltahedron can be found in [2]. Another good one is obtained by augmenting a tetrahedron in the same wayby capping off each of the four sides with another tetrahedron. Finally, any polyhedron whose faces consist of only equilateral triangles and regular hexagons can be considered to be a deltahedron by dissecting the hexagonal faces into six equilateral triangles. The simplest such polyhedron is the truncated tetrahedron, formed from a tetrahedron by cutting off four tetrahedral tips to create four equilateral triangles and four regular hexagons. Augmented polyhedra with hexagonal faces seem to be less stable than the others. That’s my method, and here’s a challenge: I wonder if anyone can create augmented toroidal deltahedra from Sonobés? I haven’t found a way to do it yet. Happy folding!

Instructions for constructing a modular version of this augmented polyhedron out of another module are given in[4], but the process is not theorized.

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